Finding correspondences

We define a distance metric using the RCS transform as the weighted L₂ error in central attribute and neighborhood function value:

$$D_\lambda({\cal R}_{{\bf I},x,y},{\cal R}_{{\bf I}',x',y'}) ~=~ (1-\lambda)\Delta N ~ + ~\lambda\Delta C$$

where the neighborhood difference is

$$\Delta N ~=~ \frac{1}{(2M_n+1)^2}\sum_{i,j} (N_{{\bf I},x,y}(i,j) - N_{{\bf I}',x',y'}(i,j))^2 .$$

The central attribute difference is similarly,

$$\Delta C ~=~ (1/a)({\bf C}_{{\bf I},x,y})^T{\bf C}_{{\bf I}',x',y'} .$$

where a is the dimension of ${\bf A}$.

The bias term $\lambda$ expresses a trade-off between the contribution of the central attribute error and the neighborhood function error. Generally the neighborhood error is the most important, since it captures the spatial structure at the given point. However, in certain cases of spatial ambiguity the central attribute value is critical for making the correct match unambiguous. For example in the image shown in Figure 2(c), the neighborhood component of the RCS transform would be roughly equal for the marked point and a point located just below the top lip (centered in the dark region of the open mouth). A modest value of $\lambda$ disambiguates this case.

To perform a correspondence search given a point (x,y) in an image ${\bf I}$, we compute the RCS transform ${\cal R}_* = {\cal R}_{{\bf I},x,y}$ and search for the point $(\hat{x},\hat{y})$ in a second image ${\bf I}'$ such that

$$(\hat{x},\hat{y}) = \arg\min_{x',y'\in W_{x,y}} D_\lambda ( {\cal R}_*, {\cal R}_{{\bf I}',x',y'} )$$

where W_x,y is a search window of radius M_w centered at (x,y).